To solve the equation \(^{n}P_{3} - 6(^{n}C_{4}) = 0\), we start by using the formulas for permutations and combinations:

\(^{n}P_{r} = \frac{n!}{(n-r)!}\)
\(^{n}C_{r} = \frac{n!}{r!(n-r)!}\)

### Step 1: Substitute the formulas

For \(^{n}P_{3}\):

\(^{n}P_{3} = \frac{n!}{(n-3)!}\)

For \(^{n}C_{4}\):

\(^{n}C_{4} = \frac{n!}{4!(n-4)!}\)

### Step 2: Substitute into the equation

The equation becomes:

\(\frac{n!}{(n-3)!} - 6 \left(\frac{n!}{4!(n-4)!}\right) = 0\)

### Step 3: Factor out \(n!\)

Factoring \(n!\) from both terms gives:

\(n! \left(\frac{1}{(n-3)!} - \frac{6}{4!(n-4)!}\right) = 0\)

Since \(n! \neq 0\), we can simplify to:

\(\frac{1}{(n-3)!} - \frac{6}{4!(n-4)!} = 0\)

### Step 4: Simplify the second term

Recall that \(4! = 24\), so:

\(\frac{6}{4!(n-4)!} = \frac{6}{24(n-4)!} = \frac{1}{4(n-4)!}\)

### Step 5: Rewrite the equation

The equation now is:

\(\frac{1}{(n-3)!} - \frac{1}{4(n-4)!} = 0\)

### Step 6: Multiply through by \(4(n-3)!(n-4)!\)

Multiplying through by \(4(n-3)!(n-4)!\) gives:

\(4(n-4)! - (n-3)! = 0\)

### Step 7: Rewrite \((n-3)!\)

Recall that \((n-3)! = (n-3)(n-4)!\), so:

\(4(n-4)! - (n-3)(n-4)! = 0\)

Step 8: Factor out \((n-4)!\)

Factoring out \((n-4)!\) gives:

\((n-4)! \left(4 - (n-3)\right) = 0\)

Since \((n-4)! \neq 0\), we have:

\(4 - (n-3) = 0\)

Solving for \(n\):

\(n - 3 = 4 \implies n = 7\)

The value of \(n\) is:

\(\boxed{7}\)